HIV is a topic that has been greatly discussed and researched due to its impact on human population. Many campaigns have been put into place, and people have been made aware of the various effects of the disease. This paper considers a fractional-order HIV epidemic model with the inclusion of prostitution in the population and its consequences on the disease transmission. The model describes the spread of HIV disease in a system consisting of a population of susceptible males and the female sex workers. The focus is on the spread of HIV by female sex workers through prostitution, because in the present world sexual transmission is the major cause of the HIV transmission. The fractional derivatives are taken in Caputo sense and the numerical solution of the model is obtained by L1 scheme which involves the memory trace that can capture and integrate all past activity. Positivity and boundedness of the solution and the stability conditions of the fractional-order system are determined. Moreover, model equilibria are determined, and their stability analysis are considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle's invariant principle. The threshold quantity namely basic reproduction number ${\mathcal{R}}_0$ is calculated and analyzed for the disease status. On the basis of ${\mathcal{R}}_0$, the disease progress can be determined i.e., the population is free from the disease if ${\mathcal{R}}_0<1$ and disease spreads in the population if ${\mathcal{R}}_0>1$. Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Further, numerical simulations of the model and their graphical presentations are performed to visualize the effectiveness of our theoretical results and to observe the impact of the arbitrary order derivative. The results obtained show the effectiveness and strength of the applied L1 scheme.

由于艾滋病毒对人类的影响，艾滋病毒已经成为人们广泛讨论和研究的话题。已经开展了许多运动，人们已经意识到这种疾病的各种影响。本文考虑了人口中包括卖淫及其对疾病传播的影响的分数阶艾滋病毒流行模型。该模型描述了艾滋病病毒在由易感男性和女性性工作者组成的系统中的传播。重点是女性性工作者通过卖淫传播艾滋病毒，因为在当今世界，性传播是艾滋病毒传播的主要原因。分数导数是在Caputo意义上获取的，并且该模型的数值解是通过L1方案获得的，该方案涉及可以捕获和整合所有过去活动的记忆轨迹。确定了该溶液的正性和有界性以及分数阶系统的稳定性条件。此外，确定了模型平衡，并使用分数Routh-Hurwitz稳定性准则和分数La-Salle不变原理考虑了其稳定性分析。阈值数量即基本再现数量 并使用分数Routh-Hurwitz稳定性准则和分数La-Salle不变原理来考虑其稳定性分析。阈值数量即基本再现数量 并使用分数Routh-Hurwitz稳定性准则和分数La-Salle不变原理来考虑其稳定性分析。阈值数量即基本再现数量${\mathcal{[R}}_0$计算并分析疾病状态。在...的基础上${\mathcal{[R}}_0$，可以确定疾病的进展，即如果 ${\mathcal{[R}}_0<\mathrm{1个}$ 疾病在人群中传播 ${\mathcal{[R}}_0>\mathrm{1个}$。同时，通过使用Lyapunov函数方法，讨论了地方平衡点的全局动力学。此外，进行了模型及其图形表示的数值模拟，以可视化我们理论结果的有效性并观察任意阶导数的影响。获得的结果表明了所应用的L1方案的有效性和强度。